Statistics - Correlation Co-efficient


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Correlation Co-efficient

A correlation coefficient is a statistical measure of the degree to which changes to the value of one variable predict change to the value of another. In positively correlated variables, the value increases or decreases in tandem. In negatively correlated variables, the value of one increases as the value of the other decreases.

Correlation coefficients are expressed as values between +1 and -1.

A coefficient of +1 indicates a perfect positive correlation: A change in the value of one variable will predict a change in the same direction in the second variable.

A coefficient of -1 indicates a perfect negative: A change in the value of one variable predicts a change in the opposite direction in the second variable. Lesser degrees of correlation are expressed as non-zero decimals. A coefficient of zero indicates there is no discernable relationship between fluctuations of the variables.

Formula

${r = \frac{N \sum xy - (\sum x)(\sum y)}{\sqrt{[N\sum x^2 - (\sum x)^2][N\sum y^2 - (\sum y)^2]}} }$

Where −

  • ${N}$ = Number of pairs of scores

  • ${\sum xy}$ = Sum of products of paired scores.

  • ${\sum x}$ = Sum of x scores.

  • ${\sum y}$ = Sum of y scores.

  • ${\sum x^2}$ = Sum of squared x scores.

  • ${\sum y^2}$ = Sum of squared y scores.

Example

Problem Statement:

Calculate the correlation co-efficient of the following:

XY
12
35
45
48

Solution:

${ \sum xy = (1)(2) + (3)(5) + (4)(5) + (4)(8) = 69 \\[7pt] \sum x = 1 + 3 + 4 + 4 = 12 \\[7pt] \sum y = 2 + 5 + 5 + 8 = 20 \\[7pt] \sum x^2 = 1^2 + 3^2 + 4^2 + 4^2 = 42 \\[7pt] \sum y^2 = 2^2 + 5^2 + 5^2 + 8^2 = 118 \\[7pt] r= \frac{69 - \frac{(12)(20)}{4}}{\sqrt{(42 - \frac{(12)^2}{4})(118-\frac{(20)^2}{4}}} \\[7pt] = .866 }$


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